Microscopic traffic flow model

Microscopic traffic flow models are a class of scientific models of vehicular traffic dynamics.

In contrast, to macroscopic models, microscopic traffic flow models simulate single vehicle-driver units, so the dynamic variables of the models represent microscopic properties like the position and velocity of single vehicles.

Car-following models

Also known as time-continuous models, all car-following models have in common that they are defined by ordinary differential equations describing the complete dynamics of the vehicles' positions ${\displaystyle x_{\alpha }}$  and velocities ${\displaystyle v_{\alpha }}$ . It is assumed that the input stimuli of the drivers are restricted to their own velocity ${\displaystyle v_{\alpha }}$ , the net distance (bumper-to-bumper distance) ${\displaystyle s_{\alpha }=x_{\alpha -1}-x_{\alpha }-\ell _{\alpha -1}}$  to the leading vehicle ${\displaystyle \alpha -1}$  (where ${\displaystyle \ell _{\alpha -1}}$  denotes the vehicle length), and the velocity ${\displaystyle v_{\alpha -1}}$  of the leading vehicle. The equation of motion of each vehicle is characterized by an acceleration function that depends on those input stimuli:

${\displaystyle {\ddot {x}}_{\alpha }(t)={\dot {v}}_{\alpha }(t)=F(v_{\alpha }(t),s_{\alpha }(t),v_{\alpha -1}(t),s_{\alpha -1}(t))}$ 

In general, the driving behavior of a single driver-vehicle unit ${\displaystyle \alpha }$  might not merely depend on the immediate leader ${\displaystyle \alpha -1}$  but on the ${\displaystyle n_{a}}$  vehicles in front. The equation of motion in this more generalized form reads:

${\displaystyle {\dot {v}}_{\alpha }(t)=f(x_{\alpha }(t),v_{\alpha }(t),x_{\alpha -1}(t),v_{\alpha -1}(t),\ldots ,x_{\alpha -n_{a}}(t),v_{\alpha -n_{a}}(t))}$ 

Examples of car-following models

• Optimal velocity model (OVM)
• Velocity difference model (VDIFF)
• Wiedemann model (1974)
• Gipps' model (Gipps, 1981)^[1]
• Intelligent driver model (IDM, 1999)^[2]
• DNN based anticipatory driving model (DDS, 2021)^[3]

Cellular automaton models

Cellular automaton (CA) models use integer variables to describe the dynamical properties of the system. The road is divided into sections of a certain length ${\displaystyle \Delta x}$  and the time is discretized to steps of ${\displaystyle \Delta t}$ . Each road section can either be occupied by a vehicle or empty and the dynamics are given by updated rules of the form:

${\displaystyle v_{\alpha }^{t+1}=f(s_{\alpha }^{t},v_{\alpha }^{t},v_{\alpha -1}^{t},\ldots )}$ 

${\displaystyle x_{\alpha }^{t+1}=x_{\alpha }^{t}+v_{\alpha }^{t+1}\Delta t}$ 

(the simulation time ${\displaystyle t}$  is measured in units of ${\displaystyle \Delta t}$  and the vehicle positions ${\displaystyle x_{\alpha }}$  in units of ${\displaystyle \Delta x}$ ).

The time scale is typically given by the reaction time of a human driver, ${\displaystyle \Delta t=1{\text{s}}}$ . With ${\displaystyle \Delta t}$  fixed, the length of the road sections determines the granularity of the model. At a complete standstill, the average road length occupied by one vehicle is approximately 7.5 meters. Setting ${\displaystyle \Delta x}$  to this value leads to a model where one vehicle always occupies exactly one section of the road and a velocity of 5 corresponds to ${\displaystyle 5\Delta x/\Delta t=135{\text{km/h}}}$ , which is then set to be the maximum velocity a driver wants to drive at. However, in such a model, the smallest possible acceleration would be ${\displaystyle \Delta x/(\Delta t)^{2}=7.5{\text{m}}/{\text{s}}^{2}}$  which is unrealistic. Therefore, many modern CA models use a finer spatial discretization, for example ${\displaystyle \Delta x=1.5{\text{m}}}$ , leading to a smallest possible acceleration of ${\displaystyle 1.5{\text{m}}/{\text{s}}^{2}}$ .

Although cellular automaton models lack the accuracy of the time-continuous car-following models, they still have the ability to reproduce a wide range of traffic phenomena. Due to the simplicity of the models, they are numerically very efficient and can be used to simulate large road networks in real-time or even faster.

Examples of cellular automaton models

• Rule 184
• Biham–Middleton–Levine traffic model
• Nagel–Schreckenberg model (NaSch, 1992)

See also

• Microsimulation

References

1. Gipps, P. G. (1981). "A behavioural car-following model for computer simulation". Transportation Research Part B: Methodological. 15 (2): 105–111. doi:10.1016/0191-2615(81)90037-0. ISSN 0191-2615. Retrieved 2022-02-17.
2. Treiber, null; Hennecke, null; Helbing, null (August 2000). "Congested traffic states in empirical observations and microscopic simulations". Physical Review E. 62 (2 Pt A): 1805–1824. arXiv:cond-mat/0002177. Bibcode:2000PhRvE..62.1805T. doi:10.1103/physreve.62.1805. ISSN 1063-651X. PMID 11088643. S2CID 1100293.
3. Isha, Most. Kaniz Fatema; Shawon, Md. Nazirul Hasan; Shamim, Md.; Shakib, Md. Nazmus; Hashem, M.M.A.; Kamal, M.A.S. (July 2021). "A DNN Based Driving Scheme for Anticipatory Car Following Using Road-Speed Profile". 2021 IEEE Intelligent Vehicles Symposium (IV). 2021 IEEE Intelligent Vehicles Symposium (IV). pp. 496–501. doi:10.1109/IV48863.2021.9575314.